Nonlinear fractional waves at elastic interfaces

We derive the nonlinear fractional surface wave equation that governs compression waves at an elastic interface that is coupled to a viscous bulk medium. The fractional character of the differential equation comes from the fact that the effective thickness of the bulk layer that is coupled to the interface is frequency dependent. The nonlinearity arises from the nonlinear dependence of the interface compressibility on the local compression, which is obtained from experimental measurements and reflects a phase transition at the interface. Numerical solutions of our nonlinear fractional theory reproduce several experimental key features of surface waves in phospholipid monolayers at the air-water interface without freely adjustable fitting parameters. In particular, the propagation distance of the surface wave abruptly increases at a threshold excitation amplitude. The wave velocity is found to be of the order of 40 cm/s in both experiments and theory and slightly increases as a function of the excitation amplitude. Nonlinear acoustic switching effects in membranes are thus shown to arise purely based on intrinsic membrane properties, namely, the presence of compressibility nonlinearities that accompany phase transitions at the interface.

Figure 1

Displacement field of the Lucassen wave, given by Eqs. (4), (8), and (9). The decay lengths in both the $x$ and $z$ directions are shown in red, with $k$ and ${\lambda }_t$ given by Eqs. (14) and (13), respectively. For the bulk medium, water is used ($\rho ={10}^3\mathrm{kg}/{\mathrm{m}}^3$ and $\eta ={10}^{-3}\mathrm{Pa}\mathrm{s}$); the interface parameters are chosen appropriately for a DPPC monolayer ($K_{2\mathrm{D}}=10\mathrm{mN}/\mathrm{m}$ and ${\rho }_{2\mathrm{D}}={10}^{-6}\mathrm{kg}/{\mathrm{m}}^2$). The shown solution has a frequency $\omega =100{\mathrm{s}}^{-1}$. Note the anisotropic scaling in the $x$ and $z$ directions and furthermore that the units in a similar plot in Ref.  [25] are wrong.

Figure 2

Langmuir isotherm and corresponding isothermal elastic modulus. The inset shows an experimentally measured pressure-area isotherm for a (Langmuir) DPPC monolayer [73]. The main plot shows the corresponding isothermal elastic modulus $K_{2\mathrm{D}}$, as calculated from Eq. (19) using the isotherm from the inset. The red dashed line shows a quadratic polynomial fit to the elastic modulus.

Figure 3

Displacement and compression profiles as a function of position $x$ for a few different fixed times. Results are shown for three different driving amplitudes $U_0^{\max }$ as indicated in the legends and for a fixed equilibrium area per lipid of $\overline{a}=88.4{\AA }^2$. Solid lines are obtained by numerical solution of the nonlinear fractional wave equation (23). Dashed lines denote analytical solutions of the linear fractional wave equation (17) with ${\rho }_{2\mathrm{D}}=0$. Compression profiles in (d)–(f) are calculated according to Eq. (28).

Figure 4

Displacement and compression profiles as a function of time $t$ for a few different fixed separations $x$ from the driven boundary. Results are shown for three different driving amplitudes $U_0^{\max }$ as indicated in the legends and for a fixed equilibrium area per lipid of $\overline{a}=88.4{\AA }^2$. Solid lines are obtained by numerical solution of the nonlinear fractional wave equation (23). Dashed lines denote analytical solutions of the linear fractional wave equation (17) with ${\rho }_{2\mathrm{D}}=0$. The dashed black curves for $x=0$ show the driving function $U_0\left(t\right)$ that is imposed as a boundary condition. Compression profiles in (d)–(f) are calculated according to Eq. (28).

Figure 5

Illustration of the time it takes to observe maximal compression at a fixed position. The blue curve shows the compression field observed at $x=8.4$ mm for a driving amplitude $U_0^{\max }={10}^{-3}$ mm. The vertical black line indicates the maximal compression $-\mathrm{\Delta }{a}^{\min }/\overline{a}$ at $x=8.4$ mm, i.e., the value of the maximum of the blue curve. The horizontal black line indicates the time difference between the boundary condition rising to $1/e$ of its maximal value, which happens at time $t_1-\tau \approx 7$ ms, and the time when the maximal compression is observed at $x=8.4$ mm, $t_{\min }\approx 20$ ms. The difference between these times is used to calculate the wave speed in Eq. (30).

Figure 6

Numerical nonlinear results. (a) The black line shows the quadratic fit to the elastic modulus shown in Fig. 2. Colored dots are the different initial areas per lipid $\overline{a}$ used for generating plots (b) and (c). (b) and (c) Numerical results for the boundary-value problem given by Eqs. (23, 24, 25, 26), with the boundary condition given by Eq. (27) and the quadratic $K_{2\mathrm{D}}$ shown in (a), for different initial areas per molecule $\overline{a}$: (b) maximal compression $-\mathrm{\Delta }{a}^{\min }/\overline{a}$ at a distance $x=8.4$ mm from the excitation source, calculated using Eq. (29) and divided by $U_0^{\max }$, and (c) the corresponding wave velocity $c$ according to Eq. (30).

Figure 7

Comparison of numerical and experimental results at a distance $x=8.4$ mm from the excitation source as a function of the excitation amplitude. The numerical data are obtained by solution of the boundary-value problem given by Eqs. (23, 24, 25, 26), with the boundary condition given by Eq. (27) and the quadratic elastic modulus $K_{2\mathrm{D}}$ shown in Fig. 2, and various driving amplitudes $U_0^{\max }$. The initial area per lipid $\overline{a}=88.4{\AA }^2$ corresponds to an initial pressure $\pi =4.3\mathrm{mN}/\mathrm{m}$ (cf. Fig. 2). Observables are calculated using Eqs. (29) and (30), with $x=8.4$ mm. The experimental data are obtained by exciting waves in a DPPC monolayer on a Langmuir trough filled with water using various driving voltages $V_0$ and measuring the FRET efficiency of pressure sensitive fluorophores at distance $x=8.4$ mm away from the excitation source [29, 73]. The equilibrium surface pressure of the DPPC monolayer is $\pi =4.3$ mN/m.